Interpolation of convex scattered data in ℝ³ using edge convex minimum L^p-norm networks, 1 < p < ∞ Free

We consider the extremal problem of interpolation of scattered data in ℝ³ by smooth curve networks with minimal L^p-norm of the second derivative for 1 < p < ∞. The problem for p = 2 was set and solved by Nielson [7]. Andersson et al. [1] gave a new proof of Nielson’s result by using a different approach. It allowed them to set and solve the constrained extremal problem of interpolation of convex scattered data in ℝ³ by minimum L²-norm networks that are convex along the edges of an associated triangulation. Partial results for the unconstrained and the constrained problems were announced without proof in [8]. The unconstrained problem for 1 < p < ∞ was fully solved in [10]. Here we present complete characterization of the solution to the constrained problem for 1 < p < ∞.